How much bandwidth can a controller tuned in a twin with an instantaneous actuator really have?

research/lag-twin

A twin with an instantaneous actuator is stable at any gain; the real loop with lag τ is stable iff τ<kd/kp=2ζ/ωn, hunts at the twin's own ωn, decays no faster than 1/(3τ) for any gains, and the real-optimal bandwidth is ≈0.26–0.46 of the limit.

8 tests

What it found

Unit mass under PD control with a first-order actuator lag τ the twin omits. The twin’s loop is stable for every gain; the real loop is stable iff τ<τ*=k_d/k_p=2ζ/ω_n (Routh, agreeing with the roots in 168 of 168 cases), hunts at exactly the twin’s own ω_n, and grows at ω_n²/(2(1+4ζ²)) per unit excess lag (formula, numeric slope and simulation agree). “Faster is better” fails: at τ=0.05, ζ=0.7 the real decay rate peaks at 4.8 (ω_n≈8) and is zero at ω_n=28 where the twin promises 19.6; the poles sum to −1/τ, so no gains decay faster than 1/(3τ). Mild lag is nearly invisible in a step test (twin RMS error ≈0.04 ε). Companion to the slew, deadband and backlash twins. Stylised: linear, one lag, noiseless.

A figure from its own output

Figure produced by research/lag-twin: overshoot — lag-twin/experiments/overshoot.svg, written by the project's own experiment code.

Paper, code and results

More in Digital twins & sim-to-real

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.